Even if they were being coerced into fighting for him, why were they always so clever and competent about it?
"Loyalty" is such a fickle thing, though, isn't it?
It is - in many respects - a rather modern idea, at least when paired with the idea of an "empire", much less a "nation-state." We think of ourselves as citizens who swell with a sense of personal pride at the idea of our country. Feels perfectly natural, right?... Well, for most of the people, through most of the time, little could have been further from the truth.
People - and by this I mean the vast, sweeping, forgotten supermajority of the population of a given region under [Enter X Authority Here] were at best subjects, and all the more often serfs (i.e. land-bound slaves by any other name). They understood "loyalty", sure, but it was a loyalty to something far closer to home than any abstract, post-Enlightenment idea of "nationhood" or "a people." It was a loyalty to family, to clan, to tribe, and to whichever god or gods one paid fealty to. Of course there was duty to one's lord or king, but that was just that - a duty, a levy, a tax to be incurred by those who must suffer what they must by those with the power to do what they will. The idea of a personal "loyalty" to a king, or an emperor, or a bureaucracy one had never seen, nor met, nor had any interaction with ever save for tax collection agents would have been so far beyond the pale as to be fantastical.
People were loyal to people - specifically the people directly in their lives - not to abstract institutions.
So, when a state of war is declared in Song China in the 13th century - the Mongols are attacking! ... of course, you will do what is required of you. You will either dutifully report to the levy your local administrator has raised to combat the barbarian incursion, or you will tend your fields to provide provisions for the troops, or whatever else might be required from the central administration to prosecute the war effort. This, far more than some sense of patriotic duty, stems from simply not wanting some hostile force sweeping through and destroying your livelihood, raping your women, and probably killing you and your sons in the process. This doubly so in the case of the Mongols, since that was their wartime modus operandi: "whoever doesn't surrender immediately, we'll annihilate... and if you do surrender immediately, maybe we won't."
So, you - perhaps you're a peasant farmer who's been conscripted into the military, or maybe you're an officer serving a Tour of Duty in the border guard to advance your career - regardless of which you are - if you're in a border guard army, you're probably not even Chinese. You're one of the so-called "cooked barbarians" (i.e. Sinicized non-Han peoples) conscripted by the throne to 以夷制夷 - yi yi zhi yi... "use barbarians to deal with barbarians." You're already in the (forced) service of a people who are not your own - and being asked to potentially lay down not only your life, but the lives of everyone and everything you love in order to defend someone else's territory. This isn't even a Chinese thing - look at the problems the Austro-Hungarian Empire faced during WWI for much the same reason... people are going to up and say "yeah, I don't love the emperor that much" and do what's best for them.
So, when your defense fails and the Mongols take whichever fortified border city you'd been assigned to, here's what they do... like clockwork: the kill everyone... everyone. Men, women, children, babies, hell there are reports of Mongol patrols sweeping cities just to kill the dogs and chickens as well. They were known to ride off having completed their extermination, only to a couple of days later send a detachment back to the city they'd just razed and slaughter any survivors who managed to hide and had since emerged and tried to bury their dead. If you lose, you're almost definitely going to have your head lopped off.
But instead of meeting this grisly fate along with the rest of your town, the khan in charge of the army (or as in the case you cite, the Great Khan himself) has for one reason or another singled you out. This was fairly common. Before the head-lopping-off extravaganza, the Mongols would typically take a sort of "survey" of the townsfolk at their mercy and attempt to see who might have skills useful to them. If you had useful skills? Congratulations, you are now in the service of the Great Khan. No ifs, ands, or buts. Don't like this idea? Well, get back in line for the execution party. I mean, those were literally the two options.
Moreover, Mongol law (called the Yassa) was extremely harsh. Once you were taken on for you skills, you had essentially a "zero strike" policy. Screw up... head chopped off. Don't do well? Head chopped off. Speak out of turn... head chopped off. You were a slave to the khan, as was everyone else both above and below you, and you lived or died at his whim. Serve him well, and he could make your life more comfortable than you ever dreamed... but mess up - even once - and all that was over with the stroke of a sword.
So here you are, a skilled archer who has shot out the horse from under the Great Khan Genghis. And then you're captured... and you're dragged kicking and screaming before the man you'd just hours ago tried - and very nearly succeeded - to kill. You're expecting to die horribly... just like everyone else around you... yet instead he not only offers you your life, but a well-paying job serving him, and a life of relative luxury for you and whichever members of your family are still breathing. Not only this, but this isn't a question of "betraying" your tribe, or your clan, or your people... this is a simple question of switching paymasters. This is a war between Mongol and Chinese - heck not even Chinese before the fall of the Liao Dynasty in the north, but the Khitan - another steppe-tribe-gone-native in China a few generations ago. You're just a vassal. Of which one? Whoever gave you the best terms... and today those terms on offer are: serve the Khannate, or lose your head.
Are you really going to be thinking about the Chinese emperor in Beijing at that point? Is the idea of "loyalty" to anything but your and yours going to cross your mind?
Here's where I'm drawing from:
Allsen, Thomas. "The rise of the Mongolian empire and Mongolian rule in north China" in The Cambridge History of China, vol. 6: Alien Regimes and Border States, 907–1368.
Davis, Richard L. "The Reign of Tu-tsung (1264–1274) and His Successors to 1279" in The Cambridge History of China Volume 5: The Sung Dynasty and its Precursors, 907–1279, Part 1.
Gernet, Jacques, (tr. H. M. Wright). "II Society: A world in process of transformation - The upper classes - The merchants - The lower classes in urban surroundings - The peasants" in Daily Life in China on the Eve of the Mongol Invasion, 1250–1276.
de Rachewiltz, Igor (tr.). The Secret History of the Mongols (元朝秘史), vol. 7.
Twitchett, Denis and Klaus-Peter Tietze. "The Liao" in The Cambridge History of China, vol. 6: Alien Regimes and Border States, 907–1368.
Can I also ask on the same level if this happened with other conquerors who gained land? Did people cooperate after being conquered? If so why? Fear or money?
With regards to the specific incident you mentioned of Genghis hiring the archer who shot his horse, you have to remember that one of the reasons Genghis was so brilliant compared to his contemporaries, especially those in Mongolia, was that he respected ability. He didn't care if you were the son of the most powerful tribal chief in the land, if you weren't fit to lead men, you didn't lead men. This example of the archer is perfect as Genghis was genuinely very impressed that the man, Jebe, had managed to do this. Genghis recognised his talent and valued it. If you imagine, as a run of the mill tribal warrior who just lost a battle, the enemy general approaches you, you having just shot his horse, a very precious thing to Mongols, you'd expect death. Why wouldn't you. For the general to not only pardon you but to offer you the chance to be an officer in his ranks because of your skill, you'd be suitably impressed. Jebe 'the arrow' rose to be a capable and loyal leader under Genghis because he valued Genghis recognising his talent and rewarding him appropriately for it, regardless of how rich, educated or important he was. Similarly, enemies whom Genghis brought into his fold experienced similar circumstances. General approaches you, you believe to laugh at how you were defeated and kill you, and then offers you a job. Not just a job, to appropriately and proudly display your skills to a general who, by virtue of being a defeated enemy, has just proved how superior he is as a leader of men. It promoted an early and strong respect for Genghis as the shock of the offer blew away preconceived prejudices. This was one of the ways in which Genghis fostered new advances in his army, one example being creating a core of cutting edge Chinese engineers to build siege weapons using this method. Also, it cannot be forgotten that Genghis killed a lot of people. A lot. This would have been a strong pull factor to join him for a lot of enemies. Then to find you've inadvertently joined the army of a military genius who rewards you justly for your individual merit? Sounds like a win to me.